- #1
drjohn15
- 13
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Jerome and Paul are competitive brothers. They live on a small farm on the northern bank of a river that runs purely east and west and that flows to the east at a rate of 1.25 m/s. The brothers have run some time trials on the farm pond, and they know that, in still water, Jerome can paddle the family canoe at a steady rate of 2.9 m/s for a considerable length of time. When Paul runs a considerable distance, it turns out that he can maintain just this same pace.
The brothers like to visit their Uncle Leo who lives on the southern bank of the river. The river is wide at this point, 1410 m across, and their uncle's dock is 170 m to the east of the point which is directly across the river from the brothers' house. Paul is not nearly as strong a paddler as is Jerome, but paddling together they can maintain a paddling speed of 3.48 m/s in the farm pond. Jerome knows that if they point their canoe due south, they will always end up to the east of Uncle Leo's dock by the time they have paddled across the river. He wants to know in which direction they should head to arrive exactly at Uncle Leo's dock without any wasted effort. Paul is finally able to determine the proper direction by using the Law of Sines, which he has learned in his high school geometry class. Make a proper drawing to express the sum of velocities for this problem, and figure out how Paul was able to determine the direction.
Law of Sines : a/sin(A) = b/ sin(B) = c/sin(C) (...or the reciprocal)
Speed when boat heading = South : ##\sqrt{3.48^2 + 1.25^2} = 3.698\ m/s##
Direction of Velocity when boat heading = South : ##\arcsin \frac{1.25}{3.698} = 19.756\deg\ E\ of\ S##
I'm not sure if I've set up the problem correctly, but I've drawn a right triangle:
start of the boat at the farm,
a vertical velocity vector directly south,
a horizontal velocity vector directly east,
and the addition of those two vectors
and the same triangle but with the distances.
I'm a little lost as to how I can use the Law of Sines to find the proper angle, and honestly I'm really not sure if I've even drawn the scenario correctly.
Any help would be greatly appreciated. Thanks!
The brothers like to visit their Uncle Leo who lives on the southern bank of the river. The river is wide at this point, 1410 m across, and their uncle's dock is 170 m to the east of the point which is directly across the river from the brothers' house. Paul is not nearly as strong a paddler as is Jerome, but paddling together they can maintain a paddling speed of 3.48 m/s in the farm pond. Jerome knows that if they point their canoe due south, they will always end up to the east of Uncle Leo's dock by the time they have paddled across the river. He wants to know in which direction they should head to arrive exactly at Uncle Leo's dock without any wasted effort. Paul is finally able to determine the proper direction by using the Law of Sines, which he has learned in his high school geometry class. Make a proper drawing to express the sum of velocities for this problem, and figure out how Paul was able to determine the direction.
Law of Sines : a/sin(A) = b/ sin(B) = c/sin(C) (...or the reciprocal)
Speed when boat heading = South : ##\sqrt{3.48^2 + 1.25^2} = 3.698\ m/s##
Direction of Velocity when boat heading = South : ##\arcsin \frac{1.25}{3.698} = 19.756\deg\ E\ of\ S##
I'm not sure if I've set up the problem correctly, but I've drawn a right triangle:
start of the boat at the farm,
a vertical velocity vector directly south,
a horizontal velocity vector directly east,
and the addition of those two vectors
and the same triangle but with the distances.
I'm a little lost as to how I can use the Law of Sines to find the proper angle, and honestly I'm really not sure if I've even drawn the scenario correctly.
Any help would be greatly appreciated. Thanks!
